On Attitude Observers and Inertial Navigation for Reference SystemFault Detection and Isolation in Dynamic Positioning

Robert H. Rogne∗, Tor A. Johansen†, Thor I. Fossen‡,Center for Autonomous Marine Operations and Systems (AMOS)

Department of Engineering CyberneticsNTNU - Norwegian University of Science and Technology

7491 Trondheim, NorwayEmail: ∗robert.rogne@itk.ntnu.no †tor.arne.johansen@itk.ntnu.no ‡thor.fossen@itk.ntnu.no

Abstract— Marine craft employing a dynamic positioningsystem rely on measurements from position reference systemsand gyrocompasses, and faults in these systems and sensorspose a serious risk of vessel drive-offs, that may in turn havedire consequences. An inertial measurement unit may provideindependent position and orientation information, provided thatthe attitude estimates of the observer in use are accurateenough. In this paper we compare three nonlinear strapdowninertial-measurement-unit-based attitude estimators aided byglobal navigation satellite systems, and put them to test in asimulation of two fault detection and isolation scenarios. Thecomparison finds that all attitude estimators under test havethe potential for use in a fault detection and isolation scheme.

I. INTRODUCTION

In ship dynamic positioning (DP) Classes 2 and 3, re-quirements dictate that the vessel must possess three inde-pendent position reference (posref) systems and have tripleredundancy in vessel sensors ([1],[2],[3]). At the open seas,the common posref systems include taut wire, hydro-acousticsystems and global navigation satellite systems (GNSS), suchas GPS, GLONASS and Galileo. The main vessel sensors aregyrocompasses and vertical reference units. These require-ments in redundancy are there to improve safety, but theirrealization is not without challenges.

Especially for the posref systems, satisfying the indepen-dence condition is not always easy. These kinds of systemsare often in the need of an external aid or reference. Forthe examples given above we have clump weights andtransponders on the sea bed (taut wire and hydro-acoustic,respectively), or satellites in space for GNSS. When thewaters are too deep, or operations carry the vessel away fromthe seabed equipment, the two former cannot be applied,leaving us only with the GNSS option. Relying on moreinstances of one type of system only can put the safetypromised by the redundancy at risk due to common failuremodes. In an incident study [4] cited by [5], “six incidentsare classified as drive-off, while five of these six incidentswere initiated due to erroneous position data from DGPSs”.

Also, in addition to the questionable redundancy, there is

This work was partly supported by the Norwegian Research Council andRolls-Royce Marine through the MAROFF program together with the Centerof Autonomous Marine Operations and Systems (AMOS) at the NorwegianUniversity of Science and Technology (grants no. 225259 and 223254).

the unavoidable question of cost. More equipment meanshigher cost, and since adding more posref systems of thesame type does not automatically enhance performance, thecost can be considered superfluous. To improve redundancy,safety, and perhaps even performance, it is therefore enticingto include another set of sensors not found in the DP classrequirements today, namely the inertial measurement unit(IMU). IMUs typically measure linear accelerations andangular velocities.

A. Inertial Measurement Units and DP

While IMUs are not in widespead use in the DP industryas of today, the idea of using an IMU in conjunction with aposref system is not new ([6],[7],[8],[9]). However, most ofthe literature focus their efforts on combining inertial sensorswith a single posref system for performance reasons. Thenotion of utilizing an IMU for improving redundancy hasbeen thought of ([10],[11]), but has had limited impact asof yet. [12] investigates using IMUs for fault detection andisolation in DP, but made some limiting assumptions on gyrobias estimation and relying on externally measured roll andpitch.

The low adoption of IMUs in DP so far could be due tothe lack of sensors with justifiable cost vs. performance ratio.Yet, with the advent of low-cost, MEMS-based sensors withever increasing performance, today’s and tomorrow’s IMUsmay enliven the industry’s enthusiasm, providing motivationfor further research on the topic.

B. Attitude estimation and IMUs

A good attitude estimate is of great importance when inte-grating the IMUs acceleration output for use in an observerfor position and velocity. To obtain high quality estimatesof the attitude, one may employ the angular velocities fromthe IMU, together with vector measurements in the inertialframe such as acceleration and compasses in a sensor fusionscheme ([13], [8], [14], [15], [16], [17]).

C. Main results

In this paper, three nonlinear state-of-the-art attitude es-timators are compared, with the motivation for usage in afault detection and isolation (FDI) setup. In [17], [16], the

observers make use of magnetometer readings as one ofthe vector measurements, while we employ a gyrocompassinstead. The contribution of this paper is related to:• Attitude observer characteristics and comparison• Stability of a cascade comprising the nonlinear ob-

servers and a translational motion observer aided byGNSS

• Simulation of two FDI cases with posref system andgyrocompass fault.

D. Organization of the paper

In Section II the system kinematics and the observerdynamics are presented. In Section III simulations are carriedout in order to evaluate the capabilities of the observers ina DP scenario with one failing posref system or one failinggyrocompass.

II. ATTITUDE OBSERVERS

A. The concept for comparison and FDI

The three attitude estimators that will be compared are twovariants of the nonlinear estimator of [16], and the observerpresented in [17]. One of them, as we shall see below, alsoprovides estimates for position and velocity, but these willonly be used internally by the observer to aid the attitudeestimation. In order to compare them for FDI in positionreference systems, a translational motion observer aided byGNSS from [18, Ch. 11] will be added in cascade to allobservers.

Fig. 1 shows the structure employed for detection andisolation of faults in either of the two posref systems, GNSS1 and 2, or either of the two gyrocompasses. The notationis explained in Section II-B. All the observers in Fig. 1 areidentical, only the inputs differ between them. A fault can beeasily be detected by comparing the measurements directly,since diverging values will indicate a fault. Isolation of thefault on the other hand will require some form of analysis ofthe observer outputs, and then tracing the fault back to thesensors feeding the observer.

For the simulation tests in this paper three such setups will

GNSS 1

GNSS 2

Gyrocompass

1

IMU

Observer

Vessel

(simulator)

Observer

Observer

Observer

p1

p2

p3

p4

pGNSS1

pGNSS2

ψGC1

aIMU

ωIMU

Gyrocompass

2

ψGC2

q1

q2

q3

q4

Fig. 1. Four observers in parallel.

be simulated in parallel, in order to compare the observerspresented in later sections.

B. General assumptions and notation

The following assumptions and simplifications are made:• Earth rotation is neglected and the North-East-Down

frame is assumed to be the inertial reference.• IMU accelerometer biases are neglected, i.e. biases are

assumed to be accounted for in calibration.• No lever arms between measuring equipment and vessel

origin.The same notation as in [18] is used:• pnm/n - position of the point om with respect to n

expressed in n• vnm/n - linear velocity of the point om with respect ton expressed in n

• ψ - yaw angle between between b and n• ωbb/n - body-fixed angular velocity• bb - gyro bias• Rn

b and Rbn = Rn

b> are the rotation matrices to n

from b and vice versa, respectively• q - quaternion representation of rotation b to n• S(x) - skew symmetric matrix:

S(x) =

0 −x3 x2

x3 0 −x1

−x2 x1 0

, x =

x1

x2

x3

• gn - local gravity vector in n• abIMU - accelerometer specific force measurement in

the body/vessel frame b• ωbIMU - angular velocity measurement of the body• pnGNSS - position measurement• ψGC - gyrocompass heading measurement

The kinematics of the system can be described as follows:

pnm/n = vnm/n (1a)

vnm/n = Rnb (q)ab + gn (1b)

q =1

2Ω(ωbb/n)q (1c)

pnGNSS = pnm/n +wp (1d)

ψgc = ψ + wψ (1e)

where

Ω(x) =

[0 −x>x −S(x)

](2)

where wp and wψ represent disturbances or measurementfaults on the position and heading, respectively. The distur-bances will change according to various faults that mightoccur in the measuring equipment.

C. Nonlinear Observer 1

1) Nonlinear Observer 1a: This observer is a furtherdevelopment of the Explicit complementary filter with biascorrection described in [14]. Observer equations from [16],sans GNSS velocity measurements and using gyrocompass

heading instead of magnetometer for the second vectormeasurement:

˙pn

m/n = vnm/n + θKpp(pnGNSS − p

nm/n) (3a)

˙vn

m/n = an + gn + θ2Kvp(pnGNSS − p

nm/n) (3b)

ξ = −Rnb (q)S(σ)abIMU+

θ3Kξp(pnGNSS − p

nm/n) (3c)

˙q =1

2Ω(ω)q (3d)

˙bb

= Proj(bb,−kI σ) (3e)

an = Rnb (q)abIMU + ξ (3f)

σ = k1vb1 ×R

nb>(q)vn1 + k2v

b2 ×R

nb>(q)vn2 (3g)

ω = ωbIMU − b+ σ (3h)

where Kpp, Kpp and Kξp are gain matrices chosen suchthat A−KC is Hurwitz, where

A =

0 I3 00 0 I3

0 0 0

, C =[I3 0 0

]K =

Kpp

Kvp

Kξp

Furthermore, θ is a tuning parameter larger than or equal to1, used to guarantee stability, kI > 0 is a gain, Proj(·, ·) isa parameter projection that puts a restriction on the estimatebb, see [15] for details. k1 and k2 of the injection term σ

are gains satisfying k1 ≥ kp and k2 ≥ kp for a kp > 0. Theother factors of (3) are:

vb1 =abIMU

||abIMU ||, vb2 =

abIMU × cb

||abIMU × cb||(4)

vn1 =an

||an||, vn2 =

an × cn

||an × cn||where

cb =

cos(ψgc)− sin(ψgc)

0

, cn =

100

(5)

The position and velocity, and attitude estimators of Nonlin-ear Observer 1 form a feedback interconnection. In [16], thisfeedback interconnection is proved to have a uniform semi-global exponentially stable (USGES) equilibrium point.

2) Nonlinear Observer 1b: From (3) we notice thatpnGNSS is part of the observer equations of Nonlinear Ob-server 1. In the case of a GNSS fault, this signal cannot betrusted, and may cause errors in the attitude estimation andFDI. We therefore also test Nonlinear Observer 1a withoutthe feedback interconnection, and call the resulting observerNonlinear Observer 1b. In this case, the feedback from theposition estimate is removed. This is achieved by instead of(4), using:

vb1 = −abIMU

||gn||, vb2 = −a

bIMU × cb

||gn × cn||(6)

vn1 =gn

||gn||, vn2 =

gn × cn

||gn × cn||

where gn = [0 0 g]>.

D. Nonlinear Observer 2

This observer is also based on the Explicit complementaryfilter with bias correction described in [14]. Observer equa-tions from [17], using gyrocompass instead of magnetometerfor the second vector measurement:

˙q =1

2Ω(ω)q (7a)

˙bb

= −kbbb

+ kbsat∆(bb)− σb (7b)

σR = k1vb1 × v

b1 + k2v

b1vb1

>(vb2 × v

b2) (7c)

σb = k3vb1 × v

b1 + k4v

b2 × v

b2 (7d)

ω = ωbIMU − b+ σR (7e)

where

vb1 = −abIMU

g, vb2 =

πvb1cb

||πvn1cn||

vn1 =

001

, vn2 =πvn

1cn

||πvn1cn||

(8)

vb1 = Rnb>(q)vn1 , v

b2 = Rn

b>(q)vn2

πx = ||x||2I3 − xx>

Furthermore, k1, k2, k3, k4, kb and ∆ are positive num-bers. sat∆(·) denotes the saturation function sat∆(x) =xmin(1,∆/||x||). k3 must be chosen to be strictly largerthan k4.

In [17], the observer (7) is proven to be “almost-globally stable" and locally exponentially stable, meaningthat [17] “for almost all initial conditions (...) the trajectory(R(t), b(t)) converges to the trajectory (R(t), b(t))", whereR = Rn

b (q) and R = Rnb (q). In total there are three initial

conditions from which the observer will not converge, eachone an unstable equilibrium point [17].

E. Translational Motion Observer aided by GNSS

The observer presented is basically the same as one foundin [18, Ch. 11] sans accelerometer bias estimation:

˙pn

m/n = vnm/n +K1(pnGNSS − pnm/n) (9a)

˙vn

m/n = Rnb (q)abimu + gn +K2(pnGNSS − p

nm/n) (9b)

This observer was also employed with success in [12]together with basic heading estimation, albeit under theassumption of known pitch and roll.

1) Stability of Nonlinear Observer and Translational Mo-tion Observer in cascade: From (1) and (9), we obtainthe following error dynamics for the Translational MotionObserver, without disturbances:

˙pn

m/n = vnm/n −K1pnm/n (10a)

˙vn

m/n = (Rnb (q)−Rn

b (q))abimu −K2pnm/n

= (I3 − Rn

b

>)Rn

b abimu −K2p

nm/n (10b)

Σ2 Σ1R~

Fig. 2. Cascade connection.

where pnm/n = pnm/n − pnm/n, vnm/n = vnm/n − v

nm/n and

Rn

b = Rnb (q)Rn

b (q)> is the error rotation matrix .Proposition 1: Let (10) and the error dynamics of Non-

linear Observer 1 or 2 be connected like in Fig. 2, with (10)as Σ1 and the Nonlinear Observer error dynamics as Σ2,the tuning parameters K1 and K2 are chosen such that thematrix

Apv :=

[−K1 I3

−K2 0

]is Hurwitz, and the tuning parameters of the NonlinearObservers satisfy the conditions found in [16] or [17]. Ifall measured signals and biases are bounded, then the originof the observer error dynamics for the cascade comprisingΣ1 and Σ2 retains the stability properties of Σ2.

Proof: Define the interconnection between Σ2 and Σ1

as h(t, Rn

b ) := (I3 − Rn

b

>)Rn

b (t)abimu(t). It can be shownthat

||h(t, Rn

b )|| = ||(I3 − Rn

b

>)Rn

b abimu||

= ||(I3 − Rn

b

>)abimu||

≤ 2amax||ε|| (11)

where amax = max(||abimu||), ε = [ε1 ε2 ε3] is the vectorpart of the quaternion q associated with R

n

b , and ||ε|| → 0as R

n

b → I3. Above we have used the properties that||Rn

b || = 1 and ||I3 − Rn

b

>|| = ||ηS(ε) − S(ε)2|| ≤ 2||ε||

[16], where η is the scalar part of q. With the conditionof (11), and making sure Apv is Hurwitz, we now have aUSGES subsystem in the case of Nonlinear Observer 1, oran “almost-globally stable" and locally exponentially stablesubsystem in the case of Nonlinear Observer 2, connected toa globally exponentially stable linear system by a linearlybounded interconnection h(t, R

n

b ). According to cascadetheory ([19]), the cascade then retains the stability propertiesof Σ2.

F. ImplementationThe nonlinear attitude observers are implemented using

the proposed implementation in [17], using exact integrationof (3d) and (7a):

qk+1 =

(cos

(T ||ωk||

2

)I4 +

T

2sinc

(T ||ωk||

2

)Ω(ωk)

)qk

where sinc(x) = sin(x)/x and T is the time step.

G. Observer comparison

The attitude observers employ many of the same princi-ples, such as fusing vector measurements with IMU gyrooutput, and putting a bound on the estimate of the gyro

bias bb

by using the saturation function or the projectionalgorithm. This is natural as they are all extensions of thework found in [14].

However, there are some major differences, the mostimportant of which are the vectors chosen for vb1 and vn1 .In Nonlinear Observer 2 ([17]), an assumption is made thatabIMU ≈ −gR

nb e3, where e3 = [0 0 1]>. This means that

pn, i.e. the vehicle’s acceleration in the inertial frame n isconsidered negligible, and that the accelerometer measuresmainly the gravity components. This is also the case forNonlinear Observer 1b. In Nonlinear Observer 1a ([16]) onthe other hand, an estimate an of pn is made with the helpof position measurements, and the estimate is fed into theinjection term σ, ideally providing more accurate informa-tion. This will become apparent if the vehicle on whichthe observers are employed is faced with rotational motionor fast translational motion, inducing Coriolis acceleration.Only the former will be an issue in a DP scenario, wherethe vessel may be exposed to waves causing the vessel tooscillate in all six degrees of freedom.

Another difference between the observers is that NonlinearObserver 2 provides a global decoupling of the roll and pitchestimations from the yaw estimations. Their motivation isthe somewhat untrustworthy properties of the magnetometerused for yaw estimation, and the magnetometer’s effect onroll and pitch without this decoupling ([17], [20]). We em-ploy a gyrocompass instead, but still it gives the opportunityto tune the yaw estimation independently of roll and pitch.This could prove valuable for FDI in gyrocompasses.

H. Fault detection and isolation

Various methods for FDI can be found in for instance [21],[22] and [23]. In this paper however, since the main focus ison the comparison of the different nonlinear observers, wewill base the FDI scheme on simple threshold values bothfor detection and identification. We will also limit ourselvesto detecting and isolating faults that appear as a semi-slowsignal drift, meaning here drifts that are slow enough to bedifficult to detect and isolate immediately, but fast enough toput the operation at risk. In our DP scenario, we define thisto be position and compass drifts below 1 m/s and 1 deg/s,respectively.

For detection of a fault, we will use the distance betweenthe two measurements we have available. When the distancereaches a certain threshold thlddet, a fault will be indicated,see Algorithm 1 for a GNSS example. When choosing thethreshold value, one must take into account normal sensordrifting, as to avoid false positives.

The next step is isolation. [12] provides a discussion onhow sensor faults will affect the equilibrium points of theobserver, and how the measurement error of the observermay be used for isolation. For example, if the GNSS ismeasuring values that differ too much from what the observeris estimating based on the IMU data, the measurement errorp = pGNSS − p will have a transient response. Here wewill monitor the outputs the Translational Motion Observers’position error for the GNSS, and the nonlinear attitude

Algorithm 1 GNSS fault detection by threshold1: x1 = pGNSS1,x, y1 = pGNSS1,y

2: x2 = pGNSS2,x, y2 = pGNSS2,y

3: r =√

(x1 − x2)2 + (y1 − y2)2

4: if r > thlddet,GNSS then5: Fault detected.

Algorithm 2 GNSS fault isolation by threshold1: for i = 1, i<=4, i++ do2: j = GNSS connected to observer i3: pi,x = pGNSSj,x − pi,x, pi,y = pGNSSj,y − pi,y4: ri =

√p2i,x + p2

i,y

5: if ri > thldiso,GNSS then6: Fault in GNSS j.

observers’ yaw (or heading) error for the gyrocompasses,and check if the error passes a given threshold thldiso, seeAlgorithm 2. The observers in the algorithm correspond tothe ones in Fig. 1.

I. Observer tuning

The main objective of the observers is the detection andisolation of faults, so gains should be chosen with that inmind, while at the same time addressing the effects of noiseand other error sources. One should choose to have a not-so-aggressive feedback, to keep the transients lingering fora while, such that the feedback does not hide the fault. Onthe other hand, setting the gains too low could hamper theestimating qualities of the observer too much, introducinglag and possibly lowering the performance to the point whereone would not be able to isolate faults at all.

With basis in the gains and parameters for the observersfound in their respective papers ([16], [17], [12]), weperformed a tuning process by trial and error. NonlinearObserver 1a turned out to be somewhat more difficult totune than the other observers. This is due to the feedbackinterconnection structure of the former, meaning that tuningthe position estimation part is affecting the attitude esti-mation, and vice versa. In Nonlinear Observer 2, we alsohave a separation of yaw from roll and pitch, simplifyingthe tuning process since the gains can be tuned somewhatindependently. After the tuning process, we opted for thefollowing tuning parameters:

1) Nonlinear Observer 1a:

k1 = 0.5, k2 = 0.5, kI = 0.03

θ = 3, Kpp = 0.6I3, Kvp = 0.11I3, Kξp = 0.006I3

and a parameter projection ensuring ||bb|| ≤ 0.51 deg/s.

2) Nonlinear Observer 1b:

k1 = 0.1, k2 = 0.5, kI = 0.03

and a parameter projection ensuring ||bb|| ≤ 0.51 deg/s.

3) Nonlinear Observer 2:

k1 = 0.1, k2 = 0.5, k3 = 0.006, k4 = 0.005

kb = 8, ∆ = 0.03

4) Translational motion observer:

K1 = 0.1I3, K2 = 0.001I3

III. SIMULATION

The simulations were carried out in MATLAB/SimulinkR2013b using a dynamic ship simulator in full 6-DOF, withwaves, vibration and measurement noise [24]. All measure-ments were set to have a zero mean normally distributednoise, except for the posref systems that have a Gauss-Markov process, see Table I for their parameters. The IMUhad a sampling frequency of 100 Hz, the GNSS 5 Hz, andthe gyrocompass 10 Hz. The observers in this paper was notpart of the feedback loop for the simulated DP system, as itis only employed for fault detection and isolation.

A. Case 1: Attitude estimation test

In the first case, a simulation of fault free operation wascarried out. No faults were simulated, as the goal of this testwas to compare the performance of the estimators.

As we can see from Fig. 3 - 8, all attitude observersmanage well, but Nonlinear Observer 1a seems to have theedge when it comes to bias, roll and pitch estimation. ForNonlinear Observer 1b and 2 in Fig. 7 some of the wavemotion seems to pollute the bias estimates on roll and pitch,relating to one of the key points of Section II-G, namely theassumptions on the vessel’s acceleration.

B. Case 2: Position reference fault

In the second case, a position reference system was setto drift at a rate of 0.4 m/s up to a maximum fault of 20meters. The fault was introduced at t = 1000 seconds. Thevessel was keeping its position at the origin. The simulationresults are shown in Figs. 9 and 10. The performance of thetranslational motion observer when it comes to estimatingthe position accurately is less than optimal, but our objectivewith this observer is solely to detect and isolate failures.

To detect the fault is trivial from Fig. 9 and since we havediverging estimates. A vertical dotted line in the figure marksthe time when a fault has been the detected by comparingGNSS measurements. From Fig. 10 showing the positionmeasurement errors of the translational motion observer, itis possible to isolate the faulty output from GNSS 1. A

TABLE IMEASUREMENT NOISES

Measurement Std. dev. Markov timeconstant

IMU acceleration 0.002 m/s2 -IMU gyro 0.08 deg/s -Position reference system x and y 1.1 m 4 minPosition reference system z 2.2 m 4 minGyrocompass 0.07 deg -

0 50 100 150 200 250 300−0.1

0

0.1

Nonlinear Observer 1a, rate biases (deg/s)Roll

0 50 100 150 200 250 300−0.1

0

0.1

Pitch

0 50 100 150 200 250 300−0.1

0

0.1

Yaw

Time (s)

EstimateReal

Fig. 3. Case 1: Nonlinear Observer 1a IMU gyro bias estimates

750 800 850 900 950 1000 1050−5

0

5

Nonlinear Observer 1a, euler angles (deg)

Roll

750 800 850 900 950 1000 1050−5

0

5

Pitch

750 800 850 900 950 1000 1050−1

0

1

2

Yaw

Time (s)

EstimateReal

Fig. 4. Case 1: Nonlinear Observer 1a attitude estimates

significant transient occurs for Observer 1b and 2 during thesensor drift-away. Translational Motion Observer 1a seemsto cancel out the position error and keep it at lower level.This is likely due to the fact that it receives its attitudefrom Nonlinear Observer 1a, which we recall has the GNSSsignal as part of the observer equations. When the attitudeis estimated with the help of the faulty GNSS signal, theattitude estimate cannot be trusted for use in FDI of that sameGNSS signal. However, the output of Observer 1a displayssome smaller transients at t = 1000 when the positionmeasurement drift starts and at t = 1050 when the driftingends.

C. Case 3: Gyrocompass fault

In the third case, a gyrocompass was set to drift at a rate of0.4 deg/s up to a maximum fault of 20 degrees. The fault wasintroduced at t = 1000 seconds. The vessel was as beforekeeping its position at the origin. The results can be seen

0 50 100 150 200 250 300−0.1

0

0.1

Nonlinear Observer 1b, rate biases (deg/s)

Roll

0 50 100 150 200 250 300−0.1

0

0.1

Pitch

0 50 100 150 200 250 300−0.1

0

0.1

Yaw

Time (s)

EstimateReal

Fig. 5. Case 1: Nonlinear Observer 1b IMU gyro bias estimates.

750 800 850 900 950 1000 1050−5

0

5

Nonlinear Observer 1b, euler angles (deg)

Roll

750 800 850 900 950 1000 1050−5

0

5

Pitch

750 800 850 900 950 1000 1050−1

0

1

2

Yaw

Time (s)

EstimateReal

Fig. 6. Case 1: Nonlinear Observer 1b attitude estimates.

in Figs. 11 and 12. As for the posref fault case, detecting afault is trivial from looking at the estimates themselves (Fig.11 as they diverge. By considering the heading estimationerror (Fig. 12), we see that we may in this case use any ofthe observers to isolate the fault with the chosen thresholds.

D. Discussion

As the simulation results demonstrate, Nonlinear Observer1a seems to have a slight edge on the others when it comesto fault free attitude estimation. This is most likely due to thefact that Nonlinear Observer 1a estimates the acceleration inthe n-frame, while the others assumes that the accelerationis negligible. In our simulations, wave excitations imposemotion on the vessel, especially in the heave direction, whichwill make significant contributions to the accelerometermeasurement abIMU .

Nevertheless, all observers were able to isolate a faulty

0 50 100 150 200 250 300−0.1

0

0.1

Nonlinear Observer 2, rate biases (deg/s)Roll

0 50 100 150 200 250 300−0.1

0

0.1

Pitch

0 50 100 150 200 250 300−0.1

0

0.1

Yaw

Time (s)

EstimateReal

Fig. 7. Case 1: Nonlinear Observer 2 IMU gyro bias estimates.

750 800 850 900 950 1000 1050−5

0

5

Nonlinear Observer 2, euler angles (deg)

Roll

750 800 850 900 950 1000 1050−5

0

5

Pitch

750 800 850 900 950 1000 1050−1

0

1

2

Yaw

Time (s)

EstimateReal

Fig. 8. Case 1: Nonlinear Observer 2 attitude estimates.

gyrocompass, and by using the attitude from the NonlinearObservers 1b and 2 as input to a Translational MotionObserver aided by GNSS, we were able to detect and isolatea drift in GNSS measurement. By doing some appropriatetuning, Nonlinear Observer 1a in the same cascade may beable to isolate the fault as well. However, that could impairthe observer’s performance and stability. In [21], there aremethods that describe applying a filter to the measurementerror, in order to generate residuals. These filters should bedesigned so that noise and other error sources are maskedout from the residual. Applying such a filter to the topmostoutput shown in Fig. 10, may yield good fault isolationresults while keeping up performance.

In [12] results very similar to Figs. 9 - 12 were obtained.However, in this current paper some of the restrictive as-sumptions in the previous work have been removed, dueto the nonlinear observers that we employ. First, we now

900 1000 1100 1200 1300 1400 1500−10

0

10

20

30

Translational Motion Observer x-position (m)

Obsv

.1a

Fault det.

900 1000 1100 1200 1300 1400 1500−10

0

10

20

30

Obsv

.1b

900 1000 1100 1200 1300 1400 1500−10

0

10

20

30

Obsv

.2

T ime (s)

FaultyNormalReal position

Fig. 9. Case 2: Translational Motion Observer position estimates and trueposition during posref fault test.

900 1000 1100 1200 1300 1400 1500−5

0

5

Translational Motion Observer x-position error (m)

Obsv

.1a

Fault threshold

900 1000 1100 1200 1300 1400 1500−5

0

5

Obsv

.1b

900 1000 1100 1200 1300 1400 1500−5

0

5

Obsv

.2

T ime (s)

FaultyNormal

Fig. 10. Case 2: Translational Motion Observer position errors duringposref fault test, where the error is the difference between the measurementand estimate.

estimate roll and pitch alongside with yaw, instead of con-sidering the two former as known external signals. Second,in [12] the IMU gyro bias was assumed to be accounted forbeforehand, and was out of scope of the paper. The observerswe use in this paper estimate the gyro bias, while still beingable to perform the FDI tasks completed in [12].

IV. CONCLUSION

In this paper we have compared three nonlinear attitudeobservers, in an attitude and bias estimation test, and intwo FDI scenarios. Nonlinear Observer 1a was found tobe slightly more suitable than Nonlinear Observer 1b and2 when it came to fault free attitude and bias estimation. Inthe first FDI test case, the nonlinear observers were pairedwith a translational motion observer aided by GNSS to detectand isolate a position fault. Nonlinear Observer 1b and 2

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0

10

20

30

Nonlinear Observer heading (deg)Obsv

.1a

Fault det.

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0

10

20

30

Obsv

.1b

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0

10

20

30

Obsv

.2

T ime (s)

FaultyNormalψ

Fig. 11. Case 3: Nonlinear Observer heading estimates and true headingduring gyrocompass fault test.

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0

1

Nonlinear Observer heading error (deg)

Obsv

.1a

Fault threshold

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Obsv

.1b

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Obsv

.2

T ime (s)

FaultyNormal

Fig. 12. Case 3: Nonlinear Observer heading error during gyrocompassfault test, where the error is the difference between the measurement andestimate.

performed well in this case, but Nonlinear Observer 1a fella bit short. In the second FDI case, the attitude observerswere used to detect and isolate a gyrocompass fault. In thiscase, all three observers managed to isolate the source of thefault.

ACKNOWLEDGMENT

The author would like to thank Torleiv Håland Bryneat the Department of Engineering Cybernetics, NTNU forproviding fruitful discussions and valuable input.

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